Partially relaxed cocoercive variational inequalities and auxiliary problem principle

Abstract

Let T:K→H be a mapping from a nonempty closed convex subset K of a finite-dimensional Hilbert space H into H. Let f:K→ℝ be proper, convex, and lower semicontinuous on K and let h:K→ℝ be continuously Frećhet-differentiable on K with h′ (gradient of h), α-strongly monotone, and β-Lipschitz continuous on K. Then the sequence {xk} generated by the general auxiliary problem principle converges to a solution x* of the variational inequality problem (VIP) described as follows: find an element x*∈K such that 〈T(x*),x−x*〉+f(x)−f(x*)≥0 for all x∈K.